In this paper, we construct a group with three real irreducible characters whose Sylow 2-subgroup is an iterated central extension of a Suzuki 2-group. This answers a question raised by Moreto and Navarro who asked whether such a group exists.
Let $Q$ be a $p$-subgroup of a finite $p$-solvable group $G$, where $p$ is a prime, and suppose that $δ$ is a linear character of $Q$ with the property that $δ(x)=δ(y)$ whenever $x,y\in Q$ are conjugate in $G$. In this situation, we show that restriction to $p$-regular elements defines a canonical bijection from the set of those irreducible ordinary characters of $G$ with Navarro vertex $(Q,δ)$ onto the set of irreducible Brauer characters of $G$ with Green vertex $Q$. Also, we use this correspondence to examine the behavior of lifts of Brauer characters with respect to normal subgroups.
For a prime $\ell$, the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with $\ell'$-degree and the corresponding set for the normalizer of a Sylow $\ell$- subgroup. Navarro's refinement suggests that the values of the characters on either side of this bijection should also be related, proposing that the bijection commutes with certain Galois automorphisms. Recently, Navarro--Spaeth--Vallejo have reduced the McKay--Navarro conjecture to certain "inductive" conditions on finite simple groups. We prove that these inductive McKay--Navarro (also c…
Let $G$ be a finite group and $Irr(G)$ the set of irreducible complex characters of $G$. Let $e_p(G)$ be the largest integer such that $p^{e_p(G)}$ divides $χ(1)$ for some $χ\in Irr(G)$. We show that $|G:\mathbf{F}(G)|_p \leq p^{k e_p(G)}$ for a constant $k$. This settles a conjecture of A. Moretó. We also study the related problems of the $p$-parts of conjugacy class sizes of finite groups.
G. Navarro has conjectured a necessary and sufficient condition for a finite group $G$ to have a self-normalizing Sylow 2-subgroup, which is given in terms of the ordinary irreducible characters of $G$. In a previous article, the author has reduced the proof of this conjecture to showing that certain related statements hold for simple groups. In this article, we describe the action of Galois automorphisms on the Howlett-Lehrer parametrization of Harish-Chandra induced characters. We use this to complete the proof of the conjecture by showing that the remaining simple groups satisfy the require…
We prove that a finite group $G$ has a normal Sylow $p$-subgroup $P$ if, and only if, every irreducible character of $G$ appearing in the permutation character $({\bf 1}_P)^G$ with multiplicity coprime to $p$ has degree coprime to $p$. This confirms a prediction by Malle and Navarro from 2012. Our proof of the above result depends on a reduction to simple groups and ultimately on a combinatorial analysis of the properties of Sylow branching coefficients for symmetric groups.
Given a prime number $p$, every irreducible character $χ$ of a finite group $G$ determines a unique conjugacy class of $p$-subgroups of $G$ which we will call the anchors of $χ$. This invariant has been considered by L. Barker in the context of finite $p$-solvable groups. Besides proving some basic properties of these anchors, we investigate the relation to other $p$-groups which can be attached to irreducible characters, such as defect groups, vertices in the sense of J. A. Green and vertices in the sense of G. Navarro.
A recent question of Gabriel Navarro asks whether it is true that the derived length of a defect group is less than or equal to the number of degrees of irreducible characters in a block. In this article, we bring new evidence towards the validity of this statement.
Let $p$ be a prime and $G$ a finite group. A complex character of $G$ is called almost $p$-rational if its values belong to a cyclotomic field $\mathbb{Q}(e^{2πi/n})$ for some $n\in \mathbb{Z}^+$ prime to $p$ or precisely divisible by $p$. We prove that, in contrast to usual $p$-rational characters, there are always "many" almost $p$-rational irreducible characters in finite groups. We obtain both explicit and asymptotic bounds for the number of almost $p$-rational irreducible characters of $G$ in terms of $p$. In fact, motivated by the McKay-Navarro conjecture, we obtain the same bound for th…
We give new evidences to the fact that the structure of a solvable group can be controlled by irreducible monomial characters. In particular we inspect the role of monomial characters in Isaacs-Navarro-Wolf's conjecture and in Gluck's conjecture.
The longstanding Alperin weight conjecture and its blockwise version have been reduced to simple groups recently by Navarro, Tiep, Spaeth and Koshitani. Thus, to prove this conjecture, it suffices to verify the corresponding inductive condition for all finite simple groups. The first is to establish an equivariant bijection between irreducible Brauer characters and weights for the universal covering groups of simple groups. Assume q is a power of some odd prime p. We first prove the blockwise Alperin weight conjecture for Sp2n(q) and odd non-defining characteristics. If the decomposition matri…
Let $p$ be a prime and $G$ a finite group. We propose a strong bound for the number of $p'$-degree irreducible characters of $G$ in terms of the commutator factor group of a Sylow $p$-subgroup of $G$. The bound arises from a recent conjecture of Navarro and Tiep [NT21] on fields of character values and a phenomenon called the continuity of $p$-rationality level of $p'$-degree characters. This continuity property in turn is predicted by the celebrated McKay-Navarro conjecture [Nav04]. We achieve both the bound and the continuity property for $p=2$.
Let $G$ be a finite $π$-separable group, where $π$ is a set of primes, and let $χ$ be an irreducible complex character that is a $π$-lift of some $π$-partial character of $G$.It was proved by Cossey and Lewis that all of the vertex pairs for $χ$ are linear and conjugate in $G$ if $2\inπ$, but the result can fail for $2\notinπ$. In this paper we introduce the notion of the twisted vertices in the case where $2\notinπ$, and establish the uniqueness for linear twisted vertices under the conditions that either $χ$ is an $\mathcal N$-lift for a $π$-chain $\mathcal N$ of $G$ or it has a linear Navar…
We verify the inductive McKay condition for simple groups of Lie type C, namely finite projective symplectic groups. This contributes to the program of a complete proof of the McKay conjecture for all finite groups via the reduction theorem of Isaacs-Malle-Navarro and the classification of finite simple groups. In an important step we use a new counting argument to determine the stabilizers of irreducible characters of a finite symplectic group in its outer automorphism group. This is completed by analogous results on characters of normalizers of Sylow d-tori in those groups.
Let $n$ and $k$ be natural numbers such that $2^k < n$. We study the restriction to $\mathfrak{S}_{n-2^k}$ of odd-degree irreducible characters of the symmetric group $\mathfrak{S}_n$. This analysis completes the study begun in [Ayyer A., Prasad A., Spallone S., Sem. Lothar. Combin. 75 (2015), Art. B75g, 13 pages] and recently developed in [Isaacs I.M., Navarro G., Olsson J.B., Tiep P.H., J. Algebra 478 (2017), 271-282].
